Degeneration of quadratic polynomial endomorphisms to a Hénon map

created by bianchi on 12 May 2018

[BibTeX]

preprint

Inserted: 12 may 2018
Last Updated: 12 may 2018

Year: 2018

ArXiv: 1803.10471 PDF

Abstract:

For an algebraic family $(f_t)$ of regular quadratic polynomial endomorphisms of $\mathbb{C}^2$ parametrized by $\mathbb{D}^*$ and degenerating to a H\'enon map at $t=0$, we study the continuous (and indeed harmonic) extendibility across $t=0$ of a potential of the bifurcation current on $\mathbb{D}^*$ with the explicit computation of the non-archimedean Lyapunov exponent associated to $(f_t)$. The individual Lyapunov exponents of $f_t$ are also investigated near $t=0$. Using $(f_t)$, we also see that any H\'enon map is accumulated by the bifurcation locus in the space of quadratic holomorphic endomorphisms of $\mathbb {P}^2$.

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